Related papers: Complex tangential flows and compactness of the $\…

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let f be a function smooth up to the boundary on a smooth bounded pseudoconvex domain D in…

We prove a version of Axler-Zheng's Theorem on smooth bounded pseudoconvex domains in C^n on which the dbar-Neumann operator is compact.

Let $\Omega\subset\mathbb{C}^m$ be a bounded pseudoconvex domain with smooth boundary. For each $k\in\mathbb{N}$, we give a sufficient condition to estimate the $\bar\partial$-Neumann operator in the Sobolev space $W^k(\Omega)$. The key…

For a domain $D$ of $\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\bar{\partial}$-Neumann problem. The paper extends to domains which are not…

We prove regularity of solutions of the $\bar\partial$-problem in the H\"older-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains…

We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness…

We study compact operators on the Bergman space of the Thullen domain defined by $\{(z_1,z_2)\in \mathbb C^2: |z_1|^{2p}+|z_2|^2<1\}$ with $p>0$ and $p\neq 1$. The domain need not be smooth nor have a transitive automorphism group. We give…

The $\bar{\partial}$-Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle, or have such…

There exist several interesting results in the literature on subnormal operator tuples having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in $\C^n$. We introduce a…

A sufficient condition for $\bar{\partial}$ to have closed range is given for pseudoconvex, possibly unbounded domains in $\mathbb{C}^n$.

We obtain weighted estimates for the $\bar{\partial}$-Neumann operator on intersections of two smooth strictly pseudoconvex domains in $\mathbb{C}^2$. The regularity estimates are described with the use of Sobolev norms with weights which…

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in $\mathbb{C}^n$ with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the…

The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other…

It is an observation due to J.J. Kohn that for a smooth bounded pseudoconvex domain D in $C^n$ there exists s>0 such that the dbar-Neumann operator on D maps $W^s_{(0,1)}(D)$ (the space of $(0,1)$-forms with coefficient functions in…

In this paper we deal with the following question: is it true that any bounded smooth pseudoconvex domain in $\mathbb{C}^n$ whose boundary contains a $q$-dimensional complex manifold $M$ necessarily has a noncompact…

Let M of real dimension 2n-1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in C^N (n less than or equal to N), of codimension one or more in C^N, and endowed with the induced CR structure. We…

The $L^2$ theory of the $\bar\partial$ operator on domains in $\mathbb{C}^n$ is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate but a family of basic estimates that we call a family of…

By using a variant Property $(P_q)$ of Catlin, we discuss the relation of small set of weakly pseudoconvex points on the boundary of pseudoconvex domain and compactness of the $\overline{\partial}$-Neumann operator. In particular, we show…

On a smooth, bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^n$, to verify that Catlin's Property ($P$) holds for $b\Omega$, it suffices to check that it holds on the set of D'Angelo infinite type boundary points. In this note, we…

Let X be a Hermitian complex space of pure dimension n. We show that the d-bar-Neumann operator on (p,q)-forms is compact at isolated singularities of X if q>0 and p+q is not equal to n-1 or n. The main step is the construction of compact…